Hi, I'm Professor Don Mueller although I'm perhaps better known as Doctor Bones. Today, we're going to talk about how you can use linear algebra in chemistry. In this case we're going to be balancing chemical reactions using linear algebra. The system I'm going to start with is the chemical reaction with NH3,ammonia, plus O2, oxygen forming N2, nitrogen, and H2O, a water molecule. So we have three atomic species here. We have N, nitrogen, H, hydrogen and O, oxygen. So to set up these homogeneous equations in the matrix form, we're going to need the three atomic species and include four variables. We have four variables, in this case W, X, Y, and Z because we're going to put them in front of each of these atomic species. In this case we've got three atomic species but we have four molecular species. NH3, O2, N2, H2O, so in front of NH3 we've got W, in front of O2, X, in front of N2, Y, in front of H2O, Z. Now we can set up the equations. With respect to nitrogen we have W equals 2Y. W here meaning one nitrogen, the Y here meaning two nitrogens. So we see the relationship. W equals 2Y. We can rewrite this as W minus 2Y equals zero. Now we're going to continue this process with each of these species. So going back we've got hydrogen. Here we have hydrogen, 3W, and we've got the water molecule which has two hydrogens, 2Z. So 3W equals 2Z or 3W minus 2Z equals zero. And the final atomic species is oxygen. Going back we see we've got X, 2X for oxygen and we've got Z here because we've got one oxygen in the water molecule. So 2X equals Z. Or 2X minus Z equals zero. So these are the equations we can put in to the matrix. Finally we're into the matrix. So linear algebra. We see along the top W, X, Y, Z and this augmented part of the matrix here, N, we can put in any number we desire. Here's we're setting them equal to zero so we've got zeros in each of these columns or each of these rows. So we go across we've got one minus two, zero, zero. Three, zero, zero, minus two, zero. And finally zero, two, zero, minus one, zero. Reduced row echelon form as we run through these equations we can do the matrix mechanics. We would reduce row echelon form gives us what we're looking for. We look to the column Z, we've got minus two thirds, minus one half, minus one third and here are the final relationships for our homogeneous system. W is now two thirds Z. X is equal to one half Z. Y equals one third Z and we see the three we the two we see the three. A least common denominator for those fractions would be two times three or six so that's why we're setting Z equal to six. Once we know Z is equal to six we can go back to the original equation, W, X, Y, Z and now place those variables in. And here we are. Four ammonia molecules, three oxygen molecules react to form two nitrogen molecules plus the six water molecules. So you can do this at home. I'd like to end by saying we've got a couple of websites and here's my associate Skelly Skeleton who would like to introduce them to you. Hey Doctor Bones. You out there in TV land can go to DoctorBonesShow.com or BrainBuildingShows.com.